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An estimation procedure that is often claimed to be part of Bayesian statistics is the maximum a posteriori (MAP) estimate of an unknown quantity, that equals the mode of the posterior density with respect to some reference measure, typically the Lebesgue measure.
The EM method was modified to compute maximum a posteriori (MAP) estimates for Bayesian inference in the original paper by Dempster, Laird, and Rubin. Other methods exist to find maximum likelihood estimates, such as gradient descent, conjugate gradient, or variants of the Gauss–Newton algorithm. Unlike EM, such methods typically require the ...
where ^ is the location of a mode of the joint target density, also known as the maximum a posteriori or MAP point and is the positive definite matrix of second derivatives of the negative log joint target density at the mode = ^. Thus, the Gaussian approximation matches the value and the log-curvature of the un-normalised target density at the ...
From a given posterior distribution, various point and interval estimates can be derived, such as the maximum a posteriori (MAP) or the highest posterior density interval (HPDI). [4] But while conceptually simple, the posterior distribution is generally not tractable and therefore needs to be either analytically or numerically approximated. [5]
Many of these energy minimization problems can be approximated by solving a maximum flow problem in a graph [2] (and thus, by the max-flow min-cut theorem, define a minimal cut of the graph). Under most formulations of such problems in computer vision, the minimum energy solution corresponds to the maximum a posteriori estimate of a solution.
Variational Bayes can be seen as an extension of the expectation–maximization (EM) algorithm from maximum likelihood (ML) or maximum a posteriori (MAP) estimation of the single most probable value of each parameter to fully Bayesian estimation which computes (an approximation to) the entire posterior distribution of the parameters and latent ...
With the uniform prior, the posterior probability distribution f(r | H = 7,T = 3) achieves its peak at r = h / (h + t) = 0.7; this value is called the maximum a posteriori (MAP) estimate of r. Also with the uniform prior, the expected value of r under the posterior distribution is
The Viterbi algorithm is a dynamic programming algorithm for obtaining the maximum a posteriori probability estimate of the most likely sequence of hidden states—called the Viterbi path—that results in a sequence of observed events.